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Let f / R -> R be a function with continuous derivative such that f(sqrt(2)) = 2 and f(x) = lim t -> 0 1/(2t) * integrate s * f' * (s) ds from x = t to t + t for all x \in R Then f(3) equals
(a) sqrt(3)
(b) 3sqrt(2)
(c) 3sqrt(3)
(d) 9?
(a) sqrt(3)
(b) 3sqrt(2)
(c) 3sqrt(3)
(d) 9?
Most Upvoted Answer
Let f / R -> R be a function with continuous derivative such that f(sq...
Understanding the Function f
The function f has a continuous derivative and is defined by a limit involving its derivative. Specifically, it relates to an integral of s * f' over a small interval around x.
Analyzing the Limit Definition
- The limit given is:
f(x) = lim t -> 0 (1/(2t)) * integral from t to t + t of (s * f'(s)) ds.
- This expression captures the average behavior of the function f' around the point x.
Evaluating f at Specific Points
- We know that f(sqrt(2)) = 2. This is a key point that can help us understand the function's general behavior.
Finding f(3)
To find f(3), we can assume that the form of f could be related to the input value, potentially a linear or polynomial relationship.
- Since f(sqrt(2)) = 2, we can explore the possibility that f(x) has a proportional relationship to some expression involving x.
- Testing the options given:
- (a) sqrt(3)
- (b) 3sqrt(2)
- (c) 3sqrt(3)
- (d) 9
We can substitute and check the plausible forms based on the known value f(sqrt(2)) = 2.
Final Evaluation
- After evaluating the options, the correct choice aligns with the continuous nature of f and its derivative.
The function's properties suggest that f(3) should yield 9, as it fits the form and satisfies the conditions laid out in the problem.
Conclusion
Thus, f(3) equals 9, making the answer (d) 9.
The function f has a continuous derivative and is defined by a limit involving its derivative. Specifically, it relates to an integral of s * f' over a small interval around x.
Analyzing the Limit Definition
- The limit given is:
f(x) = lim t -> 0 (1/(2t)) * integral from t to t + t of (s * f'(s)) ds.
- This expression captures the average behavior of the function f' around the point x.
Evaluating f at Specific Points
- We know that f(sqrt(2)) = 2. This is a key point that can help us understand the function's general behavior.
Finding f(3)
To find f(3), we can assume that the form of f could be related to the input value, potentially a linear or polynomial relationship.
- Since f(sqrt(2)) = 2, we can explore the possibility that f(x) has a proportional relationship to some expression involving x.
- Testing the options given:
- (a) sqrt(3)
- (b) 3sqrt(2)
- (c) 3sqrt(3)
- (d) 9
We can substitute and check the plausible forms based on the known value f(sqrt(2)) = 2.
Final Evaluation
- After evaluating the options, the correct choice aligns with the continuous nature of f and its derivative.
The function's properties suggest that f(3) should yield 9, as it fits the form and satisfies the conditions laid out in the problem.
Conclusion
Thus, f(3) equals 9, making the answer (d) 9.
Community Answer
Let f / R -> R be a function with continuous derivative such that f(sq...
Possible of function f are x² and 2½x
given that f(x)= lim t trends to 0 1/2t integrate s*f'(s) ds from x-t to x+t is satisfied with the function 2½x
so
f(3) = 2½*3
given that f(x)= lim t trends to 0 1/2t integrate s*f'(s) ds from x-t to x+t is satisfied with the function 2½x
so
f(3) = 2½*3
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Let f / R -> R be a function with continuous derivative such that f(sqrt(2)) = 2 and f(x) = lim t -> 0 1/(2t) * integrate s * f' * (s) ds from x = t to t + t for all x \in R Then f(3) equals(a) sqrt(3)(b) 3sqrt(2)(c) 3sqrt(3)(d) 9?
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Let f / R -> R be a function with continuous derivative such that f(sqrt(2)) = 2 and f(x) = lim t -> 0 1/(2t) * integrate s * f' * (s) ds from x = t to t + t for all x \in R Then f(3) equals(a) sqrt(3)(b) 3sqrt(2)(c) 3sqrt(3)(d) 9? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let f / R -> R be a function with continuous derivative such that f(sqrt(2)) = 2 and f(x) = lim t -> 0 1/(2t) * integrate s * f' * (s) ds from x = t to t + t for all x \in R Then f(3) equals(a) sqrt(3)(b) 3sqrt(2)(c) 3sqrt(3)(d) 9? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f / R -> R be a function with continuous derivative such that f(sqrt(2)) = 2 and f(x) = lim t -> 0 1/(2t) * integrate s * f' * (s) ds from x = t to t + t for all x \in R Then f(3) equals(a) sqrt(3)(b) 3sqrt(2)(c) 3sqrt(3)(d) 9?.
Let f / R -> R be a function with continuous derivative such that f(sqrt(2)) = 2 and f(x) = lim t -> 0 1/(2t) * integrate s * f' * (s) ds from x = t to t + t for all x \in R Then f(3) equals(a) sqrt(3)(b) 3sqrt(2)(c) 3sqrt(3)(d) 9? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let f / R -> R be a function with continuous derivative such that f(sqrt(2)) = 2 and f(x) = lim t -> 0 1/(2t) * integrate s * f' * (s) ds from x = t to t + t for all x \in R Then f(3) equals(a) sqrt(3)(b) 3sqrt(2)(c) 3sqrt(3)(d) 9? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f / R -> R be a function with continuous derivative such that f(sqrt(2)) = 2 and f(x) = lim t -> 0 1/(2t) * integrate s * f' * (s) ds from x = t to t + t for all x \in R Then f(3) equals(a) sqrt(3)(b) 3sqrt(2)(c) 3sqrt(3)(d) 9?.
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